His unsupported claim that 3 million to 5 million illegal votes were cast causes the administration to hold contradictory positions that a. the 2016 election was marred by voter fraud on an unprecedented scale, and b. the president's victory in that election is 100 percent legitimate. The tension was on display repeatedly during yesterday's White House press briefing.

Trump's call for an investigation overlooks that there have been numerous inquiries into voter fraud over the past decade, and none of them have turned up evidence of a widespread problem:

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In one of the most comprehensive investigations of fraud, Justin Levitt of Loyola Law School, Los Angeles turned up 31 credible instances of voter impersonation out of more than 1 billion votes cast between 2000 and 2014. Some of those cases may have been because of clerical errors. Levitt's investigation suggests that while voter impersonation does indeed happen, it happens so rarely that the rate is approximately one instance out of ever 32 million ballots cast. This is similar to the odds of getting “heads” 25 times in a row on a coin toss.

A five-year voter fraud investigation conducted by the George W. Bush administration “turned up virtually no evidence” of organized fraud, in the words of the New York Times. While the investigation did yield 86 criminal convictions as of 2006, many of those appear to have been linked to people misunderstanding eligibility rules or filling out paperwork incorrectly.

There even have been research and investigations into voter fraud in the 2016 election. They come to similar conclusions:

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In December, a Washington Post analysis of news reports found four documented cases of voter fraud out of about 136 million votes cast. That would yield a voter fraud rate of one instance per every 34 million ballots, close to what Levitt's investigation turned up. Two of those fraud cases involved Trump voters trying to vote twice, one involved a Republican election judge trying to fill out a ballot on behalf of her dead husband, and the last involved a poll worker filling in bubbles for a mayoral candidate in absentee ballots in Florida.

A team of Dartmouth researchers undertook a comprehensive statistical investigation of the 2016 results, looking for evidence of abnormal voting patterns. They checked for evidence of noncitizen voting, dead people voting and tampering by election officials. They didn't find any. “Our findings do strongly suggest, however, that voter fraud concerns fomented by the Trump campaign are not grounded in any observable features of the 2016 presidential election,” they concluded (emphasis theirs). “There is no evidence of millions of fraudulent votes.”

Trump's assertion of widespread voter fraud contradicts statements by his campaign's lawyers, who stated unequivocally that “all available evidence suggests that the 2016 general election was not tainted by fraud or mistake.” The statement was made in a filing opposing Green Party candidate Jill Stein's recount efforts in Michigan.

In Kansas, the Republican secretary of state examined 84 million votes cast in 22 states to look for cases of duplicate registration. The project yielded 14 prosecutions, representing 0.000017 percent of the votes cast.

In 2011, Wisconsin authorities charged 20 people with fraudulent voting in the 2008 elections. Most of these were felons who were ineligible to vote.

In general, there's also a whole mountain of academic research into voter fraud, which is largely in agreement that it's essentially a nonissue, and that isolated cases that may appear to be “fraud” are often attributable to mistakes, clerical errors or carelessness.